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Finding Reproduction Numbers for Epidemic Models and Predator-Prey Models of Arbitrary Finite Dimension Using the

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This study introduces a generalized linear chain trick (GLCT) to derive general reproduction numbers for model families. This approach simplifies analysis of dynamic models like SEIRS and predator-prey systems.

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Area of Science:

  • Mathematical Biology
  • Dynamical Systems Theory
  • Epidemiology

Background:

  • Reproduction numbers are crucial for analyzing dynamic models in ecology and epidemiology.
  • Deriving reproduction numbers is typically model-specific, hindering analysis of model families with varying dimensions.

Purpose of the Study:

  • To develop a general method for deriving reproduction numbers for families of dynamic models.
  • To leverage the generalized linear chain trick (GLCT) and next generation operator for this purpose.
  • To connect these methods with continuous time Markov chains (CTMCs) and phase-type distributions.

Main Methods:

  • The generalized linear chain trick (GLCT) was combined with the next generation operator approach.
  • The GLCT was reviewed in the context of mean-field ODE models, CTMCs, and phase-type distributions.
  • The technique was applied to generalized SEIRS and predator-prey models of arbitrary finite dimensions.

Main Results:

  • General reproduction number expressions were derived for families of generalized SEIRS models.
  • General reproduction number expressions were derived for generalized finite-dimensional predator-prey models.
  • The GLCT facilitates insights by connecting ODE models with CTMC theory and phase-type distributions.

Conclusions:

  • The GLCT provides a unified framework for deriving and analyzing reproduction numbers across model families.
  • This approach simplifies the study of complex dynamical systems, including infectious disease and ecological models.
  • The integration with CTMC theory enhances the understanding of model dynamics and population behaviors.